To determine the possible number of negative real roots of the polynomial function f(x) = x5 – 2x3 + 7x2 – 2x + 2, we can use Descartes’ Rule of Signs.
According to Descartes’ Rule of Signs, the number of positive real roots can be found by counting the sign changes in f(x). However, since we are interested in negative roots, we need to evaluate f(-x).
f(-x) = (-x)5 – 2(-x)3 + 7(-x)2 – 2(-x) + 2
= -x5 + 2x3 + 7x2 + 2x + 2.
Now, we will look at the signs of the terms in f(-x):
-1 for x5 (negative)
+2 for x3 (positive)
+7 for x2 (positive)
+2 for x (positive)
+2 for the constant (positive)
From these signs, we observe the following changes in sign:
- From -1 to +2 (1 change)
- No change from +2 to +7
- No change from +7 to +2
- No change from +2 to +2
Thus, there is 1 sign change in the expression for f(-x), which indicates that there is exactly one negative real root for the function f(x).
In conclusion, the possible number of negative real roots of the function is 1.